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# Bidirectional reflectance distribution function

The bidirectional reflectance distribution function (BRDF; $\left\{f_r\left(omega_i , omega_o\right) \right\}$) is a 4-dimensional function that defines how light is reflected at an opaque surface. The function takes an incoming light direction, $omega_i$, and outgoing direction, $omega_o$, both defined with respect to the surface normal $n$, and returns the ratio of reflected radiance exiting along $omega_o$ to the irradiance incident on the surface from direction $omega_i$. Note that each direction $omega$ is itself parameterized by azimuth angle $heta$ and elevation $phi$, therefore the BRDF as a whole is 4-dimensional. The BRDF has units sr-1, with steradians (sr) being a unit of solid angle.

Definition

The BRDF was first defined by Edward Nicodemus in the mid-sixtiesCite journal
volume = 4
issue = 7
pages = 767–775
last = Nicodemus
first = Fred
title = Directional reflectance and emissivity of an opaque surface
journal = Applied Optics
url = http://ao.osa.org/abstract.cfm?id=13818
format = abstract
] . The modern definition is:

$f_r\left(omega_i, omega_o\right)=frac\left\{dL_r\left(omega_o\right)\right\}\left\{dE_i\left(omega_i\right)\right\}=frac\left\{dL_r\left(omega_o\right)\right\}\left\{L_i\left(omega_i\right)cos\left( heta_i\right)domega_i\right\}$

where $L$ is the radiance, $E$ is the irradiance, and $heta_i$ is the angle made between $omega_i$ and the surface normal, $n$.

Physically based BRDFs

Physically based BRDFs have additional properties, including,

* obeying Helmholtz reciprocity: $f_r\left(omega_i , omega_o\right) = f_r\left(omega_o , omega_i\right)$.
* conserving energy: $forall omega_i, int_Omega f_r\left(omega_i, omega_o\right),cos\left\{ heta_o\right\} domega_o le 1$

Applications

The BRDF is a fundamental radiometric concept, and accordingly is used in computer graphics for photorealistic rendering of synthetic scenes (see the Rendering equation), as well as in computer vision for many inverse problems such as object recognition.

Models

BRDFs can be measured directly from real objects using calibrated cameras and lightsources [cite web |author=Rusinkiewicz, S. |title=A Survey of BRDF Representation for Computer Graphics |accessdate=2007-09-05 |url=http://www.cs.princeton.edu/~smr/cs348c-97/surveypaper.html] ; however, many phenomenological and analytic models have been proposed including the Lambertian reflectance model frequently assumed in computer graphics. Some useful features of recent models include:

* accommodating anisotropic reflection
* editable using a small number of intuitive parameters
* accounting for Fresnel effects at grazing angles
* being well-suited to Monte Carlo methods.

ome examples

* Lambertian model, representing perfectly diffuse (matte) surfaces by a constant BRDF.
* Phong reflectance model, a phenomenological model akin to plastic-like specularity. [B. T. Phong, Illumination for computer generated pictures, Communications of ACM 18 (1975), no. 6, 311–317.]
* Blinn-Phong model, resembling Phong, but allowing for certain quantities to be interpolated, reducing computational overhead. [cite journal | journal = Proc. 4th annual conference on computer graphics and interactive techniques | title = Models of light reflection for computer synthesized pictures | author = James F. Blinn | date = 1977 | url = http://portal.acm.org/citation.cfm?doid=563858.563893 | doi = 10.1145/563858.563893 | pages = 192]
* Torrance-Sparrow model, a general model representing surfaces as distributions of perfectly-specular microfacets.K. Torrance and E. Sparrow. Theory for Off-Specular Reflection from Roughened Surfaces. J. Optical Soc. America, vol. 57. 1976. pp. 1105–1114.]
* Cook-Torrance model, a specular-microfacet model (Torrance-Sparrow) accounting for wavelength and thus color shifting. [R. Cook and K. Torrance. "A reflectance model for computer graphics". Computer Graphics (SIGGRAPH '81 Proceedings), Vol. 15, No. 3, July 1981, pp. 301–316.]
* Ward's anisotropic model, a specular-microfacet model with a elliptical-Gaussian distribution function dependent on surface tangent orientation (in addition to surface normal).cite conference
first = Gregory J.
last = Ward
title = Measuring and modeling anisotropic reflection
booktitle = Proceedings of SIGGRAPH
pages = 265–272
year = 1992
doi = 10.1145/133994.134078
accessdate = 2008-02-03
]
* Oren–Nayar model, a "directed-diffuse" microfacet model, with perfectly-diffuse (rather than specular) microfacets. [S.K. Nayar and M. Oren, " [http://www1.cs.columbia.edu/CAVE/publications/pdfs/Nayar_IJCV95.pdf Generalization of the Lambertian Model and Implications for Machine Vision] ". International Journal on Computer Vision, Vol. 14, No. 3, pp. 227–251, Apr, 1995]
* Ashikhmin-Shirley model, allowing for anisotropic reflectance, along with a diffuse substrate under a specular surface. [Michael Ashikhmin, Peter Shirley, An Anisotropic Phong BRDF Model, Journal of Graphics Tools 2000]
* HTSG (He,Torrance,Sillion,Greenberg), a comprehensive physically-based model. [X. He, K. Torrance, F. Sillon, and D. Greenberg, A comprehensive physical model for light reflection, Computer Graphics 25 (1991), no. Annual Conference Series, 175–186.]
* Fitted Lafortune model, a generalization of Phong with multiple specular lobes, and intended for parametric fits of measured data. [E. Lafortune, S. Foo, K. Torrance, and D. Greenberg, Non-linear approximation of reflectance functions. In Turner Whitted, editor, SIGGRAPH 97 Conference Proceedings, Annual Conference Series, pp. 117–126. ACM SIGGRAPH, Addison Wesley, August 1997.]

Acquisition

Traditionally, BRDF measurements were taken for a specific lighting and viewing direction at a time using gonioreflectometers. Unfortunately, using such a device to densely measure the BRDF is very time consuming. One of the first improvements on these techniques used a half-silvered mirror and a digital camera to take many BRDF samples of a planar target at once. Since this work, many researchers have developed other devices for efficiently acquiring BRDFs from real world samples, and it remains an active area of research.

ee also

*BSDF
*Photometry (astronomy)
*Reflectance
*Albedo
*Opposition spike
*Gonioreflectometer
*Specular highlight

*Cite book
edition = 1
publisher = Springer
isbn = 3540430970
pages = 756
last = Lubin
first = Dan
coauthors = Robert Massom
title = Polar Remote Sensing: Volume I: Atmosphere and Oceans
date = 2006-02-10

*Cite book
edition = 1
publisher = Morgan Kauffmann
isbn = 012553180X
pages = 1019
last = Matt
first = Pharr
coauthors = Greg Humphreys
title = Physically Based Rendering
date = 2004

*Cite journal
volume = 103
issue = 1
pages = 27–42
last = Schaepman-Strub
first = G.
coauthors = M.E. Schaepman, T.H. Painter, S. Dangel, J.V. Martonchik
title = Reflectance quantities in optical remote sensing--definitions and case studies
journal = Remote Sensing of Environment
accessdate = 2007-10-18
date = 2006-07-15
url = http://www.sciencedirect.com/science/article/B6V6V-4K427VX-1/2/d8f9855bc59ae8233e2ee9b111252701
doi = 10.1016/j.rse.2006.03.002

References

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